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Table of Contents
Acknowledgements; Contents; Part I Preliminaries; 1 Introduction; 2 Quantum Field Theory Set Up; References; 3 Combinatorial Classes and Rooted Trees; 3.1 Combinatorial Classes and Augmented Generating Functions; 3.2 Combinatorial Specifications and Combinatorial Dyson-Schwinger Equations; References; 4 The Connes-Kreimer Hopf Algebra; 4.1 Combinatorial Hopf Algebras; 4.2 The Connes-Kreimer Hopf Algebra of Rooted Trees; 4.3 Physical Properties; 4.4 Abstract Properties; References; 5 Feynman Graphs; 5.1 Half Edge Graphs; 5.2 Combinatorial Physical Theories; 5.3 Renormalization Hopf Algebras.
5.4 Insertion and the Invariant Charge5.5 Graph Theory Tools; 5.6 Feynman Rules; References; Part II Dyson-Schwinger Equations; 6 Introduction to Dyson-Schwinger Equations; References; 7 Sub-Hopf Algebras from Dyson-Schwinger Equations; 7.1 Simple Tree Classes Which Are Sub-Hopf; 7.2 More Physical Situations; References; 8 Tree Factorial and Leading Log Toys; References; 9 Chord Diagram Expansions; 9.1 Converting the Dyson-Schwinger Equation to Differential Form; 9.2 Rooted Connected Chord Diagrams; 9.3 The s=2, k=1 Result; 9.4 Binary Trees and the General Result; References.
10 Differential Equations and the (Next-To)m Leading Log Expansion10.1 The (Next-To)m Leading Log Expansions; 10.2 Combinatorial Expansions of the Log Expansions; References; Part III Feynman Periods; 11 Feynman Integrals and Feynman Periods; References; 12 Period Preserving Graph Symmetries; 12.1 Planar Duality: Fourier Transform; 12.2 Completion; 12.3 Schnetz Twist; 12.4 Products and Subdivergences; References; 13 An Invariant with These Symmetries; References; 14 Weight; 14.1 Denominator Reduction; 14.2 Weight Drop and Double Triangles; References; 15 The c2 Invariant; References.
16 Combinatorial Aspects of Some Integration AlgorithmsReferences; Index.
5.4 Insertion and the Invariant Charge5.5 Graph Theory Tools; 5.6 Feynman Rules; References; Part II Dyson-Schwinger Equations; 6 Introduction to Dyson-Schwinger Equations; References; 7 Sub-Hopf Algebras from Dyson-Schwinger Equations; 7.1 Simple Tree Classes Which Are Sub-Hopf; 7.2 More Physical Situations; References; 8 Tree Factorial and Leading Log Toys; References; 9 Chord Diagram Expansions; 9.1 Converting the Dyson-Schwinger Equation to Differential Form; 9.2 Rooted Connected Chord Diagrams; 9.3 The s=2, k=1 Result; 9.4 Binary Trees and the General Result; References.
10 Differential Equations and the (Next-To)m Leading Log Expansion10.1 The (Next-To)m Leading Log Expansions; 10.2 Combinatorial Expansions of the Log Expansions; References; Part III Feynman Periods; 11 Feynman Integrals and Feynman Periods; References; 12 Period Preserving Graph Symmetries; 12.1 Planar Duality: Fourier Transform; 12.2 Completion; 12.3 Schnetz Twist; 12.4 Products and Subdivergences; References; 13 An Invariant with These Symmetries; References; 14 Weight; 14.1 Denominator Reduction; 14.2 Weight Drop and Double Triangles; References; 15 The c2 Invariant; References.
16 Combinatorial Aspects of Some Integration AlgorithmsReferences; Index.